European Baccalaureate Mathematics (5-period course) cheat sheet

For a quadratic ax² + bx + c = 0, the discriminant is Δ = b² − 4ac; two real roots if Δ > 0, one double root if Δ = 0, no real roots if Δ < 0.

Sum and product of roots of ax² + bx + c = 0: x₁ + x₂ = −b/a and x₁x₂ = c/a.

The binomial expansion of (a + b)ⁿ has general term C(n, k)·aⁿ⁻ᵏ·bᵏ, where C(n, k) = n!/(k!(n−k)!).

Laws of logarithms: log(xy) = log x + log y, log(x/y) = log x − log y, log(xᵖ) = p·log x; valid only for x, y > 0.

Change of base: log_b(x) = ln x / ln b; also log_b(x) = 1 / log_x(b).

Laws of indices: aᵐ·aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁻ⁿ = 1/aⁿ, a^(m/n) = ⁿ√(aᵐ).

Solving an inequality by multiplying or dividing by a negative number reverses the inequality sign.

For |x − a| < r the solution set is the open interval (a − r, a + r); for |x − a| > r it is the union of two rays.

Exponential equation b^x = c is solved as x = log(c)/log(b), requiring b > 0, b ≠ 1, c > 0.

Domain of a function: exclude values making a denominator zero and values making an even-index radicand negative; log arguments must be strictly positive.

Composition (f ∘ g)(x) = f(g(x)) is defined only when x lies in the domain of g and g(x) lies in the domain of f.

Inverse function f⁻¹ exists iff f is injective on its domain; the graph of f⁻¹ is the reflection of the graph of f in the line y = x.

Transformations: y = f(x − h) + k shifts h right and k up; y = a·f(bx) scales vertically by a and horizontally by 1/b.

Even function: f(−x) = f(x), graph symmetric about the y-axis. Odd function: f(−x) = −f(x), graph symmetric about the origin.

A rational function has a vertical asymptote where the denominator is zero and the numerator is not; horizontal asymptotes come from limits at ±∞.

Sign table of a quotient or product changes sign at each simple zero of numerator or denominator, giving the solution set of an inequality.

Exponential f(x) = aˣ with a > 1 is strictly increasing; logarithmic f(x) = log_a x is its inverse and has vertical asymptote x = 0.

lim_{x→a} f(x) exists iff the left-hand and right-hand limits both exist and are equal; the value f(a) itself is irrelevant.

Standard limits: lim_{x→0} (sin x)/x = 1, lim_{x→0} (eˣ − 1)/x = 1, lim_{x→0} (ln(1 + x))/x = 1.

Derivative rules: (uv)′ = u′v + uv′; (u/v)′ = (u′v − uv′)/v²; (f ∘ g)′ = f′(g(x))·g′(x).

d/dx of sin x, cos x, tan x, eˣ, ln x are cos x, −sin x, 1/cos²x, eˣ, 1/x respectively.

f is increasing on an interval where f′ > 0 and decreasing where f′ < 0; a sign change of f′ at a critical point gives a local extremum.

f″ > 0 means convex (concave up); a point where f″ changes sign is an inflection point.

Fundamental theorem: ∫ₐᵇ f(x) dx = F(b) − F(a) where F′ = f.

Integration by parts: ∫u dv = uv − ∫v du; by substitution: ∫f(g(x))g′(x) dx = ∫f(u) du with u = g(x).

The area between curves y = f(x) and y = g(x) from a to b is ∫ₐᵇ |f(x) − g(x)| dx.

Dot product: u·v = |u||v|cos θ = u₁v₁ + u₂v₂ + u₃v₃; u ⊥ v iff u·v = 0.

Magnitude |u| = √(u₁² + u₂² + u₃²); unit vector in direction u is u/|u|.

Cross product in 3D gives a vector perpendicular to both; its magnitude |u × v| equals the area of the parallelogram spanned by u and v.

Line in space: r = a + t·d where a is a point on the line and d a direction vector; parallel lines have proportional direction vectors.

Angle between two lines is found from cos θ = |d₁·d₂|/(|d₁||d₂|); the absolute value gives the acute angle.

Distance from point P to plane ax + by + cz + d = 0 is |ax₀ + by₀ + cz₀ + d|/√(a² + b² + c²).

Plane equation n·(r − a) = 0, i.e. ax + by + cz = k, where n = (a, b, c) is a normal vector.

Volume of parallelepiped spanned by u, v, w is the absolute value of the scalar triple product u·(v × w).

Pythagorean identity sin²x + cos²x = 1; also 1 + tan²x = 1/cos²x.

Addition formulas: sin(a ± b) = sin a cos b ± cos a sin b; cos(a ± b) = cos a cos b ∓ sin a sin b.

Double angle: sin 2x = 2 sin x cos x; cos 2x = cos²x − sin²x = 1 − 2sin²x = 2cos²x − 1.

Sine rule: a/sin A = b/sin B = c/sin C = 2R, where R is the circumradius.

Cosine rule: a² = b² + c² − 2bc·cos A; used when two sides and the included angle, or all three sides, are known.

Area of triangle = ½ab·sin C; also = ½ · base · height.

General solutions: sin x = sin α gives x = α + 2kπ or x = π − α + 2kπ; cos x = cos α gives x = ±α + 2kπ.

Tangent has period π and is undefined at x = π/2 + kπ; sine and cosine have period 2π.

For y = A sin(Bx + C) + D: amplitude |A|, period 2π/|B|, phase shift −C/B, vertical shift D.

P(A ∪ B) = P(A) + P(B) − P(A ∩ B); for mutually exclusive events the last term is zero.

Conditional probability P(A|B) = P(A ∩ B)/P(B); independence means P(A ∩ B) = P(A)P(B).

Bayes' theorem: P(A|B) = P(B|A)P(A) / P(B), with P(B) expanded over a partition of the sample space.

Binomial distribution: P(X = k) = C(n, k)pᵏ(1 − p)ⁿ⁻ᵏ, mean np, variance np(1 − p).

For a discrete random variable, E(X) = Σ xᵢpᵢ and Var(X) = E(X²) − [E(X)]².

Standardising: z = (x − μ)/σ; for a normal distribution about 68% lies within μ ± σ and about 95% within μ ± 2σ.

Mean of grouped data uses class midpoints: x̄ = Σfᵢxᵢ / Σfᵢ; median is the value with cumulative relative frequency 0.5.

Correlation coefficient r lies in [−1, 1]; r near 0 indicates weak linear association, and correlation does not imply causation.

Regression line of y on x passes through (x̄, ȳ) and has slope b = Cov(x, y)/Var(x).

Arithmetic sequence: uₙ = u₁ + (n − 1)d; sum Sₙ = n(u₁ + uₙ)/2 = n(2u₁ + (n − 1)d)/2.

Geometric sequence: uₙ = u₁·rⁿ⁻¹; sum Sₙ = u₁(1 − rⁿ)/(1 − r) for r ≠ 1.

Sum to infinity of a geometric series exists iff |r| < 1, and then S = u₁/(1 − r).

Proof by induction: establish the base case, assume the statement for n = k, then derive it for n = k + 1.

Σ_{k=1}^{n} k = n(n + 1)/2; Σ_{k=1}^{n} k² = n(n + 1)(2n + 1)/6.

A sequence converges to L if for every ε > 0 there is an N such that |uₙ − L| < ε for all n > N.

Recurrence uₙ₊₁ = f(uₙ) has a fixed point where u = f(u); a fixed point is attracting when |f′(u)| < 1.

Number of permutations of n distinct objects is n!; number of k-subsets is C(n, k) = n!/(k!(n − k)!).

Pigeonhole principle: placing n + 1 objects into n boxes forces at least one box to contain two or more objects.